Finite-time blowup for a complex Ginzburg-Landau equation

TL;DR

This paper proves finite-time blowup for solutions with negative energy in a complex Ginzburg-Landau equation, providing estimates of blow-up time as the parameter varies, and linking behavior to the nonlinear Schrödinger equation.

Contribution

It establishes finite-time blowup conditions and estimates for the complex Ginzburg-Landau equation, connecting the blow-up behavior to the limiting nonlinear Schrödinger equation.

Findings

Negative energy solutions blow up in finite time.

Blow-up time estimates depend on the parameter .

Behavior of blow-up time relates to the limiting Schr46dinger equation.

Abstract

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-i\theta} u_t = \Delta u+ |u|^{\alpha} u$ blow up in finite time, where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value $u(0)$, we obtain estimates of the blow-up time $T_{max}^\theta $ as $\theta \to \pm \pi /2 $. It turns out that $T_{max}^\theta $ stays bounded (respectively, goes to infinity) as $\theta \to \pm \pi /2 $ in the case where the solution of the limiting nonlinear Schr\"odinger equation blows up in finite time (respectively, is global).